Warm up A rabbit population starts with 3 rabbits and doubles every month. Write a recursive formula that models this situation. What is the number of rabbits after 6 months?

Exponential Functions

Recursions to Exponentials We can write geometric recursive formulas as exponential functions. This is nice because it allows us to find the next terms without knowing the previous.

Definition of a exponential function An exponential function is a function with the variable in the exponent. It is used to model growth and decay. The general form of the exponential function is 𝑦=𝑎 𝑏 𝑥

Look at warm up to determine what the variables mean 𝑦=𝑎 𝑏 𝑥 Let’s determine how many rabbits there are in the first 3 months. Month 0 is the starting amount. 𝑦=3 (2) 𝑥 As we can see: a= starting number b= rate of change x= number of time intervals that have passed. Month Number of rabbits 3 1 3 ∙ 2 =6 2 3 ∙ 2∙2=12 3 ∙ 2 ∙ 2∙2=24

Example 1 A house was purchased for $120,000 and is expected to increase in value at a rate of 6% per year. First write a recursive sequence modeling the value of the house. Then write an exponential function modeling the value.

Example 1: Solution A house was purchased for $120,000 and is expected to increase in value at a rate of 6% per year. Recursion: 𝑈 0 =120,000 𝑈 𝑛 =1.06 𝑈 𝑛−1 Exponential: 𝑦=120,000 1.06 𝑥

Looking at the “b” in another way: Decay: if b is less than 1 Growth: If b is greater than 1   a = initial amount before measuring growth/decay r = growth/decay rate (often a percent) x = number of time intervals that have passed

Example 2 Is the following equation modeling growth or decay? 𝑓 𝑥 =100 .3 𝑥 Decay 𝑓 𝑥 =.5 2.3 𝑥 Growth

Example 3 Sometimes we will need to determine how much growth or decay there is given values. Find the percent increase or decrease in the numbers below: 36,32

Example 3 - solution 36,32 First find the ratio between the two by doing 𝑠𝑒𝑐𝑜𝑛𝑑 𝑓𝑖𝑟𝑠𝑡 = 32 36 = .88 Since the second number is smaller than the first, we know it is decay. To find percent of decay do 1 - .88 = .11 So the percent of decrease is 11%.

Example 4 Do we do the same method if the numbers are increasing? Try and see: 63, 100.8

Example 4 Solution 63, 100.8 𝑠𝑒𝑐𝑜𝑛𝑑 𝑓𝑖𝑟𝑠𝑡 = 100.8 63 =1.6 The 1 represents the whole number, and the 6 represents the percent change, So these numbers have a 60% increase.

This is a big assignment! Do not put it off! Homework 5.1 Worksheet Problems: 1 a-c 2 4 a-d 5 a-c 6 This is a big assignment! Do not put it off!